Maximum likelihood estimation vs calculating distribution parameters “manually”

Question

I'm sorry for asking probably elementary question, but I cannot understand how estimating probability distribution parameters using maximum likelihood estimation method differs from calculating these parameters from observed data manually. For MLE we need to know the type of probability distribution anyway so why don't we just use the known formulas for calculating the corresponding parameters from observed data? I believe that MLE is somehow more general method but I cannot see what is the real advantage of MLE compared to getting these parameters "manually".

Thanks for explanation.

Tomas